along this chapter (figure V.1). We guarantee that our construction gen-erates a valid discrete gradient and that, at least in the case of surfaces, the flow of V is increasing with respect to f. Since the critical points of f : M Ñ R may not be vertices of K, |K| M (figure V.2), the no-tion of critical point is delicate in the discrete setting. The usual definino-tion is due to Banchoff [Banchoff, 1967], and relies on the Euler characteristic.
This definition can miss some essential critical points in high dimensions, and we will thus extend it naturally using either homology (section IV.2(a) Betti numbers and torsion) or directly our optimal discrete Morse function construc-tion (secconstruc-tion III.2(c) Heuristic for optimal Morse functions). The main advan-tages of this construction over other so–called discrete Morse–Smale decom-positions [Edelsbrunner et al., 2001, Edelsbrunner et al., 2003] consist in its simplicity, the rigor of the construction and the generality, since the construc-tion works on any finite cellular complex. Moreover, the construcconstruc-tion can be complemented to guarantee the position of the critical cells and their shapes, even if this step is proved not necessary for refined surfaces. Finally, Forman’s theory ensures the homology of the resulting complex, guaranteeing a consis-tent result.
95 V.1. Geometric critical points
V.3(a): Minimum. V.3(b): Maximum.
V.3(c): Regular point. V.3(d): Saddle. V.3(e): Monkey saddle.
Figure V.3: Classification of generic points on surfaces, and a non–generic saddle (e).
the Euler characteristic of its lower star differs from the one of a semi–opened disk: χplwfτq 0. The Banchoff index of a critical point is defined as this Euler characteristic:
idxpτq χplwfτq ¸
pPN
p1qpcardtσp ¡τ, f pσpq fpτqu.
Observe that this definition differs from the Morse index qpτq of a critical point τ, defined as the number of negative eigenvalues of the Hessian matrix.
However, idxpτq p1qqpτq for non–degenerated critical points, and Banchoff proved in [Banchoff, 1967] that his index is still linked to the Euler character-istic of the manifold: χpKq °
τPK0idxpτq.
Although this definition is simple and intuitive, we cannot use it to define rigorously a discrete Morse complex for the following reasons: it requires addi-tional tests even to decide if a critical point is a maximum or a minimum, and in higher dimensions, the Euler characteristic is not sufficient to capture every critical point, as it does not determine whether a complex is homeomorphic to a disk or not. Therefore, there can be some critical points essential to compute the homology properly on the Morse complex that are missed by Banchoff’s definition. In any case, our construction of discrete Morse complexes can use even an incomplete set of critical points, as it would automatically generate a complete set of critical cells from those critical points.
Figure V.4: Lower star of a saddle point.
(b) Critical points in high–dimension
We use two different definitions of critical points that coincide in all practical case. The first relies on homology computation and it is defined uniquely in all cases. If the homology of the lower star is pK,t0u,t0u. . .q, the point is regular. Otherwise, it is a maximum if β0 0, a minimum if βn1 1, a 1–saddle if β0 2 and a k–saddle if βk1 1 (figure V.5). It can also be a degenerated saddle if βk ¡1 orβ0 ¡2. In some particularly complex topological case, this definition can miss some critical points, for example if the lower star of a vertex is a homological disc. The homology is computed using the algorithm of section IV.2(a)Betti numbers and torsion.
The second definition uses our heuristic to define optimal Morse function (section III.2(c) Heuristic for optimal Morse functions). If the heuristic achieves defining a discrete gradient with no critical point on the lower star of a vertex, this vertex will be considered as regular. If not, the point will be critical, and its class is computed in the same way as in the first definition. Observe that even the lower star is not a cellular complex, the algorithm of section III.2 Greedy construction works directly. This definition is not as rigorous as the homology–
based definition, but it allows considering as critical obstructions to cancelling cells such as non–shellability, non–simple homotopy features or homological discs.
Observe that both definitions coincides with Banchoff’s one for surfaces and solids, but the class of the critical point can be computed in an easier way.
(c) From critical points to critical cells
In Forman’s discrete Morse theory, the critical elements are cells in general, instead of only points. Although this point of view gave powerful results, the detection of critical cells per se is a little more complex than
97 V.1. Geometric critical points
V.5(a): Minimum. V.5(b): Maximum.
V.5(c): 1–saddle. V.5(d): 2–saddle. V.5(e): Monkey saddle.
Figure V.5: Classification of generic critical points in solids, and a non–generic saddle (e): the link of the point is mapped onto a sphere, the blue parts represent the higher values, and the yellow ones the lower values.
detecting critical points. The main step of our algorithm actually does not require this detection, but it can be suited to control the output of the algorithm. There are two contexts to work on. In the first context, if the mesh contains more information than just the support for the geometry, for example when it has been generated to obtain some special features on the triangles, a critical cell has to be selected from the existing cells. In the second context where we are free to change the mesh, we will create a critical cell directly from a critical point.
The classification of critical points is essential for discrete Morse theory, since the index of a critical point determines the dimension of the correspond-ing critical cell. As degenerate critical points actually correspond to various critical points agglomerated, the procedure to generate a critical cell from a critical point can be repeated to identify the many critical cells of a degenerated critical point.
In both of the above contexts, the easiest part is the detection of a minimum since in Forman’s theory, a minimum is a vertex which identifies easily with a point, and our algorithm will always place the minima at the right position. A maximum τ is also easily detected, but the corresponding critical cell σ must be a cell of maximal dimension n. We will choose σ as the one containing the vertices of the link of τ having the highest value
through f. Formally, if we denote τ1 argmaxtfpτ1q, τ1 P K0Xlkτu and τi 1 argmaxtfpτ1q, τ1 PK0Xlkτiz tτj, j ¤iuu, σ is the cell incident to the first τi’s. If the complex is simplicial, we can write σspanpτ, τ1, τ2, . . . τnq.
v
w x
y
V.6(a): Maximum: fpvq ¡ fpwq ¡ fpxq ¡ fpyq and w second maximum afterv.
v
w x
y
V.6(b): Saddle: fpvq ¡ fpxq ¡ fpyq ¡fpwq and w second minimum.
Figure V.6: Construction of critical cells from a maximum and a saddle.
The construction of a saddle relies on the following observation. In the smooth case, the flow elongates a small region around a saddle in the directions where the saddle point is a local maximum. This elongation extends on both sides of the saddle point, and the lowest side is a natural direction for the flow.
The construction of a k–saddle,k ¡1, is thus similar to the construction of a maximum, but considering the argmin instead of the argmax in the definition of τi.
The construction of a 1–saddle is slightly more complex, since the lower star is not connected. For each connected componentiof the lower star, denote by σi the edge of the connected component incident to the lowest vertex. Let σ0 be the edge incident to the lowest vertex of the whole star. Then each cell σi, i0 will be critical, where σ0 is excluded because it would be the regular direction of the flow. This definition even entails degenerated saddles, such as monkey saddles.
In the second context, we can perform a vertex split operation to create the critical cell in the appropriate direction. For example a maximal point τ would be substituted by a cell of maximal dimension σ having the same combinatory as the intersection of a small ball around τ with the star of τ.
The vertices of σ (i.e. the intersections of the sphere with the edges of stτ) will be positioned all at the same height fpτq. A k–saddle is constructed in the same way, but considering a small k–cylinder instead of a ball.
99 V.2. Computation